Frequency dependence fractal structures

The third relaxation process is located in the low-frequency region and the temperature interval 50°C to 100°C. The amplitude of this process essentially decreases when the frequency increases, and the maximum of the dielectric permittivity versus temperature has almost no temperature dependence (Fig 15). Finally, the low-frequency ac-conductivity ct demonstrates an S-shape dependency with increasing temperature (Fig. 16), which is typical of percolation [2,143,154]. Note in this regard that at the lowest-frequency limit of the covered frequency band the ac-conductivity can be associated with dc-conductivity cio usually measured at a fixed frequency by traditional conductometry. The dielectric relaxation process here is due to percolation of the apparent dipole moment excitation within the developed fractal structure of the connected pores [153,154,156]. This excitation is associated with the selfdiffusion of the charge carriers in the porous net. Note that as distinct from dynamic percolation in ionic microemulsions, the percolation in porous glasses appears via the transport of the excitation through the geometrical static fractal structure of the porous medium. 

The interfacial capacitance may also be measured at solid polarizable electrodes in an impedance experiment using phase-sensitive detection. Most experiments are carried out with single crystal electrodes at which the structure of the solid electrode remains constant from experiment to experiment. Nevertheless, capacity experiments with solid electrodes suffer from the problem of frequency dispersion. This means that the experimentally observed interfacial capacity depends to some extent on the frequency used in the a.c. impedance experiment. This observation is attributed to the fact that even a single crystal electrode is not smooth on the atomic scale but has on its surface atomic level steps and other imperfections. Using the theory of fractals, one can rationalize the frequency dependence of the interfacial properties [9]. The capacitance that one would observe at a perfect single crystal without imperfections is that obtained at infinite frequency. Details regarding the analysis of impedance data obtained at solid electrodes are given in [10]. 

Electric Double Layer and Fractal Structure of Surface Electrochemical impedance spectroscopy (EIS) in a sufficiently broad frequency range is a method well suited for the determination of equilibrium and kinetic parameters (faradaic or non-faradaic) at a given applied potential. The main difficulty in the analysis of impedance spectra of solid electrodes is the frequency dispersion of the impedance values, referred to the constant phase or fractal behavior and modeled in the equivalent circuit by the so-called constant phase element (CPE) [5,15,16, 22, 35, 36]. The frequency dependence is usually attributed to the geometrical nonuniformity and the roughness of PC surfaces having fractal nature with so-called selfsimilarity or self-affinity of the structure resulting in an unusual fractal dimension... 

Conclusion. Calculations of the dependence of the conductivity and the relative permittivity of chaotic hierarchical self-similar structures of composites were performed using a fractal model in the entire range of concentrations of inhomogeneities at various frequencies of an external field. The metal-insulator transition was shown to occur not only near the percolation threshold. It was also shown that the transition depends on the concentration of the metallic phase and the frequency of the external field. 

In this subsection we will consider (distinct from the dendrimers of Sect. 8) another class of regular hyperbranched polymers. We recall that the quest for simpUcity in the study of complex systems has led to fruitful ideas. In polymers such an idea is seating, as forcefully pointed out by de Gennes [4j. Now, the price to be paid in going from linear chains to star polymers [33,194[, dendrimers [13,33,194,205] and general hyperbranched structures [216[ is that scaling (at least in its classical form) is not expected to hold anymore (at least not in a simple form, which implies power-law dependences on the frequency CO or on the time f). One of the reasons for this is that while several material classes (such as the Rouse chains) are fractal, more general structures do not necessarily behave as fractals. 

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